Applied Structural and Mechanical Vibrations 2009 Part 15 docx

In practical cases, eq (15.3), which has a general theoretical validity, modifies for two aspects. Firstly, real signals are necessary band-limited between, say, f min and f max , with outside such a frequency range. Secondly, every real system has a finite bandwidth extending from f 1 to f 2 , with f 1 =0 in the case of a DC-responsive system. Of course, f 1 and f 2 must be chosen so that and to include the signal into the system bandwidth. Therefore, eq (15.3) in practice becomes (15.4) This result points out the importance of properly tailoring the system bandwidth according to both the signal and the noise characteristics. If the noise is white or has significant components outside the signal bandwidth, it is desirable to reduce the system bandwidth as close as possible to by proper filtering, since this operation has the effect of maximizing the S/N ratio. On the other hand, keeping the system bandwidth much wider than the signal bandwidth is useless and has the only detrimental effect of collecting more noise. Unfortunately, the portion of the noise which resides within the signal bandwidth cannot be directly removed without affecting the signal as well. Special techniques can be used in these cases, such as the modulation which will be briefly presented later in this chapter. 15.3 Signal DC and AC amplification 15.3.1 The Wheatstone bridge The Wheatstone bridge represents a classical and very widespread method for measuring a small resistance variation ∆R superimposed on a much higher average value R. This situation represents a rather typical occurrence in transducers, and is for instance encountered in strain-gauge-based sensors, where ∆R/R can be as low as 1 part per million (ppm), and other resistive sensors such as resistive temperature detectors (RTD). The Wheatstone bridge consists of four resistors arranged as two resistive dividers connected in parallel to the same excitation source, as shown in Fig. 15.1. Such a source can be either constant or a function of time, and either made by a current or a voltage generator. In the following, we shall consider a constant voltage excitation V E , which is the most frequently used in practice. The bridge output voltage V o is given by: (15.5) Copyright © 2003 Taylor & Francis Group LLC A survey of the possible configurations is given in Fig. 15.2. The bridge imbalance voltage can be generally expressed as (15.7) where γ is the bridge fractional imbalance which is approximately equal to ∆ R/(4R), and exactly equal to ∆ R/(2R) and ∆ R/R in the quarter-, half- and full-bridge respectively. It can be observed that the use of tension-compression pairs increases the sensitivity over the quarter-bridge. Moreover, the nonlinearity inherent in the quarter-bridge configuration is removed since the current in each arm is constant. Another advantage of making use of the configurations incorporating multiple piezoresistors is the intrinsic temperature compensation provided. In fact, if all the strain gauges have the same characteristics and are located closely so that they experience the same temperature, their thermally induced resistance variations are equal and, as such, they do not contribute any net imbalance voltage. The same result can hardly be obtained in the quarter-bridge configuration, because the strain gauge and the Fig. 15.2 Wheatstone bridge configurations for resistive measurements: (a) quarter bridge; (b) half bridge; (c) full bridge. Copyright © 2003 Taylor & Francis Group LLC completion resistors normally have different thermal coefficients of resistance (TCR) and, moreover, are subject to different temperatures. In practical cases, the excitation voltage V E is in the range of few volts and the bridge imbalance voltage V o can be as low as few microvolts, and therefore it requires amplification. This is generally accomplished by a differential voltage amplifier, called an instrumentation amplifier (IA), with an accurately set gain typically ranging from 100 to 2000, and a very high input impedance in order not to load the bridge output by drawing any appreciable current. Since V o is proportional to V E , any fluctuation in V E directly reflects on V o causing an apparent signal. To overcome this problem, a ratiometric readout scheme is sometimes used in which the ratio is electronically produced within the signal conditioning unit, thereby providing a result which is only dependent on γ . In turn, γ is related to the input mechanical quantity to be measured through the gauge factor and the material and geometrical parameters of the elastic structure. The Wheatstone bridge can be also used with resistance potentiometers. In this case, with reference to Fig. 15.1, one side of the bridge, say the left, is made by the potentiometer so that R 1 and R 2 represents the two resistances into which the total potentiometer resistance R P is divided according to the fractional position x of the cursor. That is and with Then, the system works in the half-bridge configuration and, assuming according to eqs (15.5) and (15.7) the bridge fractional imbalance is given by The Wheatstone bridge with DC excitation may be critical in terms of S/ N ratio when the signal γ is in the low-frequency region. In fact, in this case the bandwidth of the bridge output voltage V o becomes superimposed with that of the system low-frequency noise, which is typically the largest noise component in real systems. Moreover, an additional spurious effect comes from the DC electromotive forces (EMF) arising across the junctions between different conductors present in the bridge circuit, and from their slow variation due to temperature called the thermoelectric effect. This causes a low-frequency fluctuation of the bridge imbalance indistinguishable from the signal of interest. Both problems may be greatly reduced by adopting an AC carrier modulation technique, as illustrated in the following section. 15.3.2 AC bridges and carrier modulation If reactive components have to be measured instead of resistors, such as for capacitive or inductive transducers, the bridge configuration of Fig. 15.1 can again be adopted with the resistors now substituted by the impedances Z 1 , Z 2 , Z 3 and Z 4 . Since the impedance of inductors and capacitors at DC is either zero or infinite, the bridge now requires an AC excitation, which we can assume to Copyright © 2003 Taylor & Francis Group LLC be a sinusoidal voltage expressed in complex exponential notation as An expression equivalent to eq (15.5) can then be written for the bridge output V o (t), leading to: (15.8) Similarly to the resistive bridge, the balance condition is given by which, however, involves complex impedances and hence actually implies two balance requirements, one for the magnitude and one for the phase. The balance condition is independent of the excitation amplitude V E but, in general, does depend on the frequency ω E . Equation (15.8) also describes the bridge deflection operation, with the term representing the bridge fractional imbalance γ introduced in eq (15.7) which is now a complex function of the excitation frequency. In general, both the amplitude and the phase of V o (t) depend on γ and, as such, they may vary with frequency. Therefore, the determination of γ from V o (t) for a given known excitation V E (t) can be rather involved. Fortunately, there are several cases of practical interest where the situation simplifies considerably. Suppose, for instance, that Z 1 and Z 2 represent the impedances of the two coils of an autotransformer inductive displacement transducer as described at the end of Section 14.4.3, or alternatively, the impedances of the two capacitors of a differential (push-pull) configuration used for the measurement of the seismic mass displacement in capacitive accelerometers, as mentioned in Section 14.8.4. In both cases, it can be readily shown that and where x is the fractional variation of impedance induced by the measurand around the average value Z. If the completion impedances Z 3 and Z 4 are chosen so that which is most typically accomplished by using equal resistors then γ reduces to a real number which equals x/2. In this circumstance, eq (15.8) may be rewritten avoiding the complex exponential notation with yielding (15.9) which is equivalent to the resistive half-bridge configuration. It can be noticed that the output voltage V o (t) becomes a cosinusoidal signal synchronous with Copyright © 2003 Taylor & Francis Group LLC the excitation voltage with an amplitude controlled by the bridge fractional imbalance γ . Hence, V E (t) behaves as the carrier waveform over which γ exerts an amplitude modulation. The process of extracting γ from V o (t) is called demodulation. To properly retain the sign of γ , i.e. to preserve its phase, it is necessary to make use of a so-called phase-sensitive (or coherent, or synchronous) demodulation method. In fact, if pure rectification of V o (t) were adopted then both + γ and – γ would result in the same rectified signal, thereby losing any information on the measurand sign. A typically adopted method to implement phase-sensitive demodulation employs a multiplier circuit. Such a component accepts two input voltages V M1 (t) and V M2 (t) and provides an output given by where K M is the multiplier gain factor. With reference to the block diagram of Fig. 15.3(a), the bridge output voltage is first amplified by a factor A, then is band-pass filtered around 2 ω E , for a reason that will be shortly illustrated, and then fed to one of the multiplier inputs, while the other one is connected to the excitation voltage V E (t). The multiplier output V Mo (t) is then given by (15.10) In eq (15.10) can be observed the fundamental fact that, due to the nonlinearity of the operation of multiplication, V Mo (t) includes a constant component proportional to the input signal x. The oscillating component at 2 ω E can be easily removed by low-pass filtering, and the overall output V out (t) becomes a DC voltage proportional to x given by: (15.11) To maximize accuracy, both the excitation amplitude V Em and the gains A and K M need to be kept at constant and stable values. The excitation frequency ω E is instead not critical, since it does not appear in eq (15.11). The configuration schematized in Fig. 15.3(a) for either inductive or capacitive transducers can also be adopted for resistive sensors connected in any variant of the Wheatstone bridge. Moreover, the method of AC excitation followed by phase-sensitive demodulation also represents a typical readout scheme used for LVDTs (Section 14.4.3), as illustrated in Fig. 15.3(b). In this case, for the particular transducer used, ω E is usually chosen equal to the value which zeroes the parasitic phase- shift between the voltages at the primary and the secondary at null core position. Copyright © 2003 Taylor & Francis Group LLC the signal, usually ten times greater, and the bandwidths of the band-pass and low-pass filters are properly set, then the output V out (t) reproduces the input signal without frequency distortions. 15.4 Piezoelectric transducer amplifiers 15.4.1 Voltage amplifiers In Section 14.8.1 piezoelectric accelerometers were discussed, and the equivalent electrical circuit of Fig. 14.13 was derived in which the sensor is modelled as a charge generator proportional to acceleration in parallel with the internal resistance R and capacitance C. This model generally applies to all piezoelectric transducers, such as accelerometers, force or pressure transducers, and accounts for the fact that piezoelectric sensors are self- generating. Depending on the strength of the mechanical input signal and the value of C, the voltage developed across the sensor terminals may sometimes be directly detectable by a recording instrument, such as an oscilloscope or a spectrum analyser, without any amplification. However, due to the finite internal impedance of the transducer the input impedances of the readout instrument and of the connecting cable itself generally cause significant loading of the transducer output in the case of direct connection. Therefore, the measured voltage can be considerably reduced compared to the open-circuit voltage, and the sensitivity is diminished by a factor which is neither constant nor controllable. Moreover, the direct connection is prone to interference pick-up which may significantly degrade the signal. Avoiding these effects requires voltage amplification to raise the signal level, and impedance conversion to decrease the loading by the cable and the readout instrument. This may be accomplished by making use of a voltage amplifier, whose ideal features are infinite input impedance, zero output impedance and gain G independent of frequency. Figure 15.4(a) shows the circuit diagram inclusive of the equivalent capacitances and resistances of the sensor (C, R), the sensor-to-amplifier cable the input stage of a real voltage amplifier and the amplifier-to-instrument cable plus the instrument input The voltage amplifier may be as simple as a single operational amplifier (OA) in the noninverting configuration as shown in Fig. 15.4(b) for which the gain G is equal to [5]. If G is made equal to one, it becomes a unity-gain or buffer amplifier, also called a voltage follower, since the output follows the input signal without any gain added. The voltage at the readout instrument input, in the Laplace domain, is given by (15.12) Copyright © 2003 Taylor & Francis Group LLC The ratio gives the midband gain or amplification, usually expressed in volts per picocoulomb (V/pC). It should be noticed that the amplification is critically dependent on the capacitances C S and C i . For ordinary coaxial cables C S is typically of the order of 100 pF per metre and in most cases it dominates C i . Therefore, for a given amplifier, cable type or length cannot be changed without affecting the calibration constant. For transducers based on piezoceramics this effect is less evident than with quartz, due to the fact that the internal capacitance C is generally much higher in the former case and may eventually dominate C S . The cable should be of the low-noise type, that is it must be coaxial with the outer shield devoted to blocking the radio-frequency and electromagnetic interference (RFI and EMI) and it must not suffer from the triboelectric effect. This effect consists in charge generation across the cable inner insulator due to friction when the cable is bent or twisted. Such a spurious charge appears across the cable capacitance and is directly added to the signal charge Q, therefore it may impair its detectability. The tribolectric effect can be minimized by choosing a cable of noise-free construction incorporating a lubricant layer between the insulator and the shield and, anyway, preventing cable movement by securing it in a fixed position by cable clamps or adhesive tape. The connection of an extra capacitor, sometimes called a ranging capacitor, in parallel with the amplifier input increases C T and produces a decrease in amplification that may be adjusted to scale down the sensitivity to the desired level without acting on the amplifier gain G. For a good low-frequency response the discharge time constant (DTC) must be high. A possible method would seem that of making C T very high, but this is not a good choice since it decreases the midband gain according to eq (15.13). It is better to increase R T as much as possible by choosing a high input resistance amplifier and by paying attention to any possible cause of loss of insulation in cabling and connectors, such as dirt or Fig. 15.5 Gain magnitude versus frequency for the voltage amplifier configuration. Copyright © 2003 Taylor & Francis Group LLC humidity. In the ideal case of a perfect cable and amplifier, the DTC would reduce to that intrinsic of the transducer given by RC. As voltage amplifiers, and OAs in particular, have virtually zero output impedance, to first order the presence of C o and R o causes no loading effect, as demonstrated by the fact that they do not appear in eqs (15.12) and (15.13). In practice, the output of a voltage amplifier can typically drive sufficiently long cables; however the high-frequency response drops the higher the capacitive load and, therefore, the longer the cable as qualitatively shown in Fig. 15.5. As a significant cost advantage over the use of costly low-noise cable, ordinary coaxial cable can be used at the output. In fact, the virtually zero output impedance of the amplifier shunts the cable impedance and the input impedance of the readout instrument, therefore it prevents the tribolectric charge developing a spurious voltage at the instrument terminals. Voltage amplifiers are most usually sold as in-line units that must be connected as near as possible to the transducer and, occasionally, can fit on top of its case. In the former case, it should be remembered that the length of the input cable must be kept fixed to preserve calibration. 15.4.2 Charge amplifiers The role of a charge amplifier is not that of augmenting the charge generated by the sensor, which is impossible to attain since such a charge is fixed by the strength of the mechanical input. Instead, charge amplifiers behave as charge converters which are able to transform the input charge into a voltage output through a gain factor that is virtually independent of both the sensor and the cable impedance. The circuit diagram of a charge amplifier is shown in Fig. 15.6(a). It can be noticed the presence of a voltage amplifier having a negative voltage gain –A, which is usually very high and assumed to be ideally infinite, and the parallel connection of the capacitor C f and the resistance R f which provide a feedback path from the output to the input. Again, the equivalent resistances and capacitances of the sensor, of the cables and of the input stage of the real amplifier are taken into account by inserting the corresponding lumped elements in the circuit diagram. This scheme is most often implemented in practice by making use of an OA in the inverting configuration [5], as shown in Fig. 15.6(b). By applying Kirchhoff’s current law at the amplifier input node and remembering that the current entering an ideal voltage amplifier is zero due its infinite input impedance, it can be written that (15.14) with and By considering that Copyright © 2003 Taylor & Francis Group LLC If A is made sufficiently high so that which, neglecting the resistances which are usually very high, reduces to it follows that eq (15.16) simplifies to (15.17) It can be observed that eq (15.17) is equivalent to eq (15.12) valid for a voltage amplifier. The differences are that R f and C f now replace R T and C T , the voltage gain G is absent, and the presence of the minus sign determines an inversion of the output voltage with respect to the input charge. It is important to notice that, as long as A is sufficiently high so that eq (15.16) can be replaced by eq (15.17), the voltage output is now insensitive to the sensor internal impedance, the cable impedance, and the amplifier voltage gain and input impedance. The charge-to-voltage transfer function, whose magnitude Bode plot is shown in Fig. 15.7, is only dependent on R f and C f , which are external components that may be properly chosen to set both the low-frequency limit or equivalently the DTC given by and the midband amplification –1/C f expressed in volts per picocoulomb (V/pC). The sometimes-encountered statement that charge amplifiers have a high input impedance is not correct. In fact, it is the voltage amplifier around which the charge amplifier is built that has a high input impedance. On the contrary, owing to the negative feedback, the charge amplifier actually works as a virtual short-circuit to ground, which presents an ideally zero input impedance to the transducer. In fact, for It is for this reason that a charge amplifier has the fundamental capability of bypassing the transducer and cable impedances and drawing all the generated charge Q. For signal frequencies beyond ω L such a charge is then conveyed into Cf, developing a proportional output voltage V o . The condition of a high value of the voltage gain A is usually well satisfied with OAs, which typically provide a voltage gain in the order of 10 5 at low Fig. 15.7 Gain magnitude versus frequency for the charge amplifier configuration. Copyright © 2003 Taylor & Francis Group LLC [...]... indicated as a voltage follower) and a low-output impedance RT and CT include the impedance of the transducer, of the amplifier input and of the ranging capacitor if present The product RTCT gives the system DTC, and sets the low-frequency limit With reference to eqs (15. 12) and (15. 13) with G now equal to one, for frequencies higher than ω L the output voltage Vo is given by (15. 18) The charge amplifier... cable and amplifier impedance For a charge amplifier (Section 15. 4.2) and is the DTC For constant-current internally-amplified transducers (Section 15. 4.3) with DC output coupling, eq (15. 20) is again valid, with the only difference that Vo now includes the bias voltage VB instead of being ground-referenced In the case of AC output coupling, two time constants are involved and the eq (15. 20) becomes (15. 21)... high-pass (HP), bandpass (BP) and band-reject (BR) or notch filters The gain functions |H(ω)| of each of them are graphically shown in Fig 15. 25 A simple example of a LP filter is the RC network discussed in Section 15. 4.6 with regard to its timeintegration capability In all the plots of Fig 15. 25 can be distinguished a frequency region called the passband where |H(ω)| does not vary and the signal is... Copyright © 2003 Taylor & Francis Group LLC Then, starting from Fig 15. 15(a) and assuming that the transducer side is the one that can be lifted off ground, the resulting situation is depicted in Fig 15. 15(b) Now the ground noise VG, though still present, no longer influences the signal loop and, for the readout signal Vi equals VS 15. 5.2 Inductive coupling The preceding section has pointed out that... Vi(ω) and Vo(ω) the transform of the filter input and output respectively, it holds that (15. 26) The filtering action works on the basis that, depending on the shape of H(ω), some frequencies are enhanced or passed without alteration, and others are attenuated Passing to the input and output power spectral densities SVi(ω) and SVo(ω) which can represent either signal or noise, it follows that (15. 27)... the eqs (15. 20) and (15. 21) in the Laplace domain and then antitransforming the resulting output voltage Vo(s) However, considerable insight is gained in trying to analyse and predict the time response to elementary excitation waveforms by starting from the system frequency response We have seen that the high-frequency response is affected by the combination of Ta(ω ) and the possible amplifier and mounting... reducing the S/N ratio (Section 15. 2) Therefore, the system bandwidth should be narrowed as much as possible around the signal bandwidth As an example, if you are interested in measuring the vibration amplitude of a light part oscillating at say 500 Hz, it is advisable to insert a BP filter centred on such a frequency with a suitable bandwidth to suppress wideband noise and possible interference components... both the transducer and the receiver to the common ground In the most general sense, Copyright © 2003 Taylor & Francis Group LLC Now consider the case of Fig 15. 15(a) in which the signal return path is provided by a second wire, again of resistance Rc, connected between G1 and G2 This is a very common situation in practice, and it is often thought that grounding both the source and the receiver at... practical in each application It should be noticed that the same operation cannot be made on the circuit of Fig 15. 14, since the ground conductor is an integral part of the signal loop and breaking it would interrupt the return path Fig 15. 15 Two-wire voltage transmission between transducer and readout unit (a) Ground noise VG affects the readout voltage Vi due to ground-loop problem (b) Ground loop... transducer, and by 2 if present Such time constants introduce a high-pass filtering action and the system is not responsive to DC acceleration If only the DTC 1 is present, at the overall gain is attenuated by –3 dB with respect to its midband value, and it decreases at a 20 dB/decade (or 6 dB/octave) rate for The becomes π /4 at ω1, and tends to phase shift is π /2 at low frequency zero for If both 1 and . eq (15. 16) simplifies to (15. 17) It can be observed that eq (15. 17) is equivalent to eq (15. 12) valid for a voltage amplifier. The differences are that R f and C f now replace R T and. times greater, and the bandwidths of the band-pass and low-pass filters are properly set, then the output V out (t) reproduces the input signal without frequency distortions. 15. 4 Piezoelectric. the fractional position x of the cursor. That is and with Then, the system works in the half-bridge configuration and, assuming according to eqs (15. 5) and (15. 7) the bridge fractional imbalance is